Since $H$ is strictly convex, it has a unique maximizer, call it $p_{max}$. A discrete random variable has a discrete uniform distribution if each value of the random variable is equally likely and the values of the random variable are uniformly distributed throughout some specified interval.. Now compare that to a normal distribution, say $N(\mu, \sigma^2)$. When I discussed it with John von Neumann, he had a better idea. Among probability distributions which are nonzero over a finite range of values , the maximum-entropy distribution is the uniform distribution. In this example, the maximally uncertain distribution is given by the uniform distribution over past states, i.e. a characteristic of a dice. The variation $\delta f(x)$ refers to a rate of change of $f(x)$ with respect to "time". THE ENTROPY OF THE NORMAL DISTRIBUTION INTRODUCTION The "normal distribution" or "Gaussian distribution" or Gaussian probability density function is defined by N(x; m, s) = 1 (2ps2)1/2 e-(x-m)2/2s2. A planet you can take off from, but never land back. apply to documents without the need to be rewritten? &=\left(\frac1{f(x)}f(x)+\log(f(x))\right)\delta f(x)\\ Note that in $(3)$, $-(1+\log(f(x)))\to\infty$ as $f(x)\to0$. I'm confused. For continuous distributions, normal distribution corresponds to maximum entropy. Also, wikipedia has a brief discussion on this as well: wiki. To handle varying functions, we will make use of the Calculus of Variations. Standard uniform distribution: If a =0 and b=1 then the resulting function is called a standard unifrom distribution. Site design / logo 2022 Stack Exchange Inc; user contributions licensed under CC BY-SA. In fact, I started writing the answer quite differently, aiming to show that you'd got the entropy wrong! More precisely, consider the unit simplex $\Delta_n=\{(p_1,\dots,p_n): p_i\ge 0,\sum_i p_i=1\}$.Then $H$ may be considered a function $H: \Delta_n\to \mathbb{R}$, and it is easy to show that it is strictly convex. The one will have less information content and more weight, the others more information content and less weight. There's clearly more information here. It only takes a minute to sign up. I asked you "Dude, where's my car?" . Theorem 5.1 states, that the continous probability density on [a,b] with = a + b 2 that maximizes entropy is uniform distribution q ( x) = 1 b a, x [ a, b]. 1+\log(f(x))=c_0\cdot\color{#C00}{1}\tag5 2022 Coursera Inc. All rights reserved. The distribution is represented by U (a, b). This illustrate the fact that for a given number $n$ of symbols, entropy per symbol is maximal at $\log_2 n\,$bit/symbol when all the symbols have the same probability, that is uniform distribution. It then follows, since entropy is maximized at \end{align} The below diagram shows. For example, a fair dice has entropy $H(X)=6\,\left(\frac16\log_26\right)=\log_26\approx2.585\ldots\,$bit/symbol. \delta(\log(f(x))f(x)) *log (frequency./binWidth (nz))) It seems that the most common references (i.e. And if that second bit is 1, Then the weather in Gotham City is rainy. Deutsche Bahn Regional. The distributions package contains parameterizable probability distributions and sampling functions. Let's plot the entropy and visually confirm that p=0.5 gives the maximum. Should I avoid attending certain conferences? However, if you told me "I saw your car one hour ago on Route 66 heading from Washington, DC" - this is not a uniform distribution anymore. By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. Number of unique permutations of a 3x3x3 cube. We present a class of integer sequences fc n g with the property that for every p-invariant and ergodic positive-entropy measure on T, fc n x (mod 1)g is uniformly distributed for-almost every x. $$ The best answers are voted up and rise to the top, Not the answer you're looking for? DB Bus. Can you see therefore how a uniform distribution is less ordered than others? Assuming the symbols can take values $x_i$, are produced independently at random (that is, earlier outcomes have no influence on later ones), and each symbol $x_i$ has fixed probability $p(x_i)\ge0$ with $1=\sum p(x_i)$, then the uniform distribution p u (X t1), . $$, $$ So let's tackle that. A good measure of uncertainty achieves its highest values for uniform distributions. Nice one, thanks. if both integrals exist. Is this meat that I was told was brisket in Barcelona the same as U.S. brisket? This doesn't cause a problem in $(2)$ since $-\log(f(x))f(x)$ is bounded by $\frac1e$. The entropy of $\{p_1 + \varepsilon, p_2 -\varepsilon,p_3,,p_n\}$ minus the entropy of $\{p_1,p_2,p_3,,p_n\}$ equals, $$-p_1\log\left(\frac{p_1+\varepsilon}{p_1}\right)-\varepsilon\log(p_1+\varepsilon)-p_2\log\left(\frac{p_2-\varepsilon}{p_2}\right)+\varepsilon\log(p_2-\varepsilon)$$ A special case of this family is the Logistic-Uniform distribution. How many axis of symmetry of the cube are there? Do we ever see a hobbit use their natural ability to disappear? Movie about scientist trying to find evidence of soul, How to rotate object faces using UV coordinate displacement. If it does, remember to upvote! Who wants to go skiing? Abstract The evolution of element distribution during laser cladding involves two dynamic behaviors, i.e., liquid molten pool flow and FeCoCrNi high-entropy alloy (HEA) coatings solidification. Do we ever see a hobbit use their natural ability to disappear? Uniform probability yields maximum uncertainty and therefore maximum entropy. Connect and share knowledge within a single location that is structured and easy to search. for a random variable $X \in$ set $A$ :- $H(X)= \sum_{x_i \in A} -p(x_i) \log (p(x_i)) $. $$ Moreover, by twice differentiating this expression you can check that the sufficient conditions for the LM method holds. (+1). i.e: they answer "similarly" if you switch "yes" and "no" ($kid_2$ is like $kid_1$ if $kid_1$ woke up on the left foot). Execution plan - reading more records than in table. Joint differential entropy of sum of random variables: $h(X,X+Y)=h(X,Y)$? Stack Overflow for Teams is moving to its own domain! Asking for help, clarification, or responding to other answers. frequency = counts (nz)/sum (counts (nz)); H = -sum (frequency. So which means that maximisation always is with respect to constraints ? \bbox[5px,border:2px solid #C0A000]{f(x)=\frac1{b-a}}\tag6 There are many misconceptions of entropy, so you're in good company. The uniform distribution is generally used if you want your desired results to range between the two numbers. What do you call an episode that is not closely related to the main plot? On the other hand, for practical adversaries, a Cryptographically Secure PRNG that outputs integers symbols in $[0,n)$ is indistinguishable from a true RNG with entropy of $\log_2 n\,$bit per symbol and the same output domain. In this example, let's use the weather information in Gotham City as the random phenomenon. Return Variable Number Of Attributes From XML As Comma Separated Values, Find all pivots that the simplex algorithm visited, i.e., the intermediate solutions, using Python. How to print the current filename with a function defined in another file? Great, I'll upvote you once I have some reputation. Can plants use Light from Aurora Borealis to Photosynthesize? -\int_a^b\log(f(x))f(x)\,\mathrm{d}x\tag2 Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. Language eh? Welcome to CV. It's messing up most of the answers here. $$ Read and process file content line by line with expl3. So in our next video, we'll look at a case where they're not equally probable or when the probability's distribution is no longer uniform. H &= -\sum_{i=0}^{n-1} p_i \log p_i - (1-q)\log q\\ So, weather, In Gotham City. According to Wikipedia, the uniform distribution is the "maximum entropy probability distribution". Maximization is always performed subject to constraints on the possible solution. We will find a new probability density with higher entropy. Unlike a probability, a probability density function can take on values greater than one; for example, the uniform distribution on the interval [0, ] has probability . It also states that "multivariate distribution with max- imum entropy, for a given covariance, is a Gaussian". For example, we could also take $a = 0$ and $b = 1/2$, giving entropy $-\ln(2) < 0$, where as in the discrete case entropy is always non-negative. The pulse function is a key to deriving the unit hydrograph theory. The purpose of entropy metric is to measure the amount of information. The density of the maximum entropy distribution for this class is constant on each of the intervals [a j-1,a j). How does DNS work when it comes to addresses after slash? Uniform Distribution. Therefore, the distribution is often abbreviated U, where U stands for uniform distribution. However, the independence property tells us that this relationship should hold: What is rate of emission of heat from a body at space? Is it possible for a gas fired boiler to consume more energy when heating intermitently versus having heating at all times? On the whole real line there is no uniform probability density. $$ If X is a discrete random variable with distribution given by . $$, $$ If entropy is high, we should consider that disorder is high too, but in a uniform distribution we don't have disorder because all items have the same chance to appear. Another way of saying "discrete uniform distribution" would be "a known, finite . where p(x) log(1/p(x)) is understood to be zero whenever p(x) = 0.. ;). For example, we could also take $a = 0$ and $b = 1/2$, giving entropy $-\ln(2) < 0$, where as in the discrete case entropy is always non-negative. Use MathJax to format equations. :). nz = counts>0; % Index to non-zero bins. And if the second bit is 1 given the first bit is 1, then the weather condition is Gotham City is cloudy. Suppose the $p_j$ are not all equal, say $p_1 < p_2$. It has a mean value of $\mu$. If the weather, if the bit is 0, then we know that it's either going to be sunny or rainy but we don't know which one. It tells the expected/average number of bits necessary to encode the value of a symbol, knowing the characteristics of the source. How to go about finding a Thesis advisor for Master degree, Prove If a b (mod n) and c d (mod n), then a + c b + d (mod n). If we wish to maximize $(2)$ for all distributions satisfying $(1)$, we need to find all $f$ so that $(2)$ is stationary; that is, $\delta$ of the integral in $(2)$ vanishes: Does baro altitude from ADSB represent height above ground level or height above mean sea level? This module studies information entropy to quantify randomness. Browse other questions tagged, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, This question is answered (in passing) at. I do not understand it at all. Your answer is "it's somewhere in USA between Atlantic and Pacific Oceans." And that person wants to communicate the weather in Gotham City to an outsider who doesn't know the weather condition in Gotham City. Browse other questions tagged, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company. -\int_a^b(1+\log(f(x))\,\delta f(x)\,\mathrm{d}x=0\tag3 Discrete Uniform Distribution. Then there is no uncertainty. To complete the proof, we want to show this is positive for small enough $\varepsilon$. Therefore our Lemma says $h(p)\leq h(q)$, with equality if and only if $p$ is uniform. I did the discrete case, then was going to say "and the continuous case follows similarly", but thought that I'd just do it anyway as it's easy. There are a few simple ways to overcome the problem in $(3)$. Yes, we're constructing a measure for the lack of information so we want to assign its highest value to the least informative distribution. Yes, there is! The di When the constraints are that all probability must vanish beyond predefined limits, the maximum entropy solution is uniform. Covalent and Ionic bonds with Semi-metals, Is an athlete's heart rate after exercise greater than a non-athlete. I'm confused. We have Note that $p(x_i)$ define a discrete probability distribution, so their sum is 1. Thanks for contributing an answer to Mathematics Stack Exchange! The entropy of a uniform distribution is l n ( b a). Entropy satisfies the criterion. Concealing One's Identity from the Public When Purchasing a Home. Is it possible for a gas fired boiler to consume more energy when heating intermitently versus having heating at all times? Consider the discrete distribution case. So now we need to send another bit. 1, pp. Lognormal Distribution as Maximum Entropy Probability Distribution, Kullback-Leibler divergence WITHOUT information theory, Proving that Shannon entropy is maximised for the uniform distribution. How can I write this using fewer variables? In uniform distribution the entropy is high. f (x) = 1/ (max - min) Here, min = minimum x and max = maximum x 2. \begin{align} Putting all of this together leads to the definition of the Shannon entropy: For a chance variable $X$ taking values in $\mathcal X$ and distributed according to $P_X$, the Shannon entropy of $X$ is $$H_b(X) = \Sigma_{x \in supp(P_x)}P_X[x]*\log_b\frac{1}{P_X[x]} = \mathop{\mathbb{E}}_X[-\log_b P_X(X)] $$ Where, $supp(P_X) = \{x \in \mathcal X: P_X[x] > 0 \}$. How come there is no uncertainty? $$-\int_I p\log p dx\leq -\int_I p\log q dx$$ To show this, we must maximize the entropy, lg with respect to , subject to the constraints I did the discrete case, then was going to say "and the continuous case follows similarly", but thought that I'd just do it anyway as it's easy. So just to recap, H is m times log of capital N. Where small m is the number of independent events. Entropy (S) is a state function that can be related to the number of microstates for a system (the number of ways the system can be arranged) and to the ratio of reversible heat to kelvin temperature.It may be interpreted as a measure of the dispersal or distribution of matter and/or energy in a system, and it is often described as representing the "disorder" of the system. $$ Based on this example which kid's answer will have most uncertainty? It seems the answer is no. And those are the four possible weather conditions in Gotham City. Similarly if the second bit is 0 given that the first bit is 1, then the weather in Gotham City is snowy. Nonetheless, it can serve as a criterion for measuring how far/close a distribution is to the uniform distribution. Difference between mutual and conditional information. Why do "nothing up my sleeve numbers" have low entropy? And the answer to this question, will it yield the information entropy? \end{align} The pattern that emerges from our experiment is that broad distributions have the highest . Entropy is a continuous function of the $n$-tuples $(p_1,,p_n)$, and these points lie in a compact subset of $\mathbb{R}^n$, so there is an $n$-tuple where entropy is maximized. Covariant derivative vs Ordinary derivative. How is this statement valid? Recalling that $\log(1 + x) = x + O(x^2)$ for small $x$, the above equation is Mobile app infrastructure being decommissioned. What mathematical algebra explains sequence of circular shifts on rows and columns of a matrix? This is an example of the uniform distribution. \begin{align} $$\log_2 x = {\ln x \over \ln 2}.$$, Hope this helps! \end{align} Site design / logo 2022 Stack Exchange Inc; user contributions licensed under CC BY-SA. The entropy for our three example distributions is 0 (Dirac delta), 174 (Gaussian), and 431 (uniform). If X is a continuous random variable with probability density p(x), then the entropy of X is defined as . $\Delta_n=\{(p_1,\dots,p_n): p_i\ge 0,\sum_i p_i=1\}$, $\sigma p=(p_{\sigma(1)},\dots, p_{\sigma(n)})$, $$ So the weather information of one day is completely independent or will not affect the weather condition in another day. Which probability distribution has maximum entropy ? That is, no maximizing function will have $f(x)=0$. If entropy is high, we should consider that disorder is high too, but in a uniform distribution we don't have disorder because all items have the same chance to appear. Why is HIV associated with weight loss/being underweight? p u (X t = X t *): = . If one bit is sent, and I suppose if that bit is 0 then we know that the weather is sunny or rainy.. Then, if that first bit is 1, then we know that it's going to be neither sunny nor rainy. $$, $$ With $a=0$ and $b=1$ this reduces to zero. Shannon's Entropy defines axiomatic characterization of a discrete probability density function P P which gives event i probability pi p i. H (p1,p2,p3 . -\int_a^b\log(f(x))f(x)\,\mathrm{d}x\tag2 Denote $q = 1-\sum_{i=0}^{n-1} p_i$. So how much information does a random phenomenon contain? So those two bits provide the information of Gotham City. $$ Then the sum is equal to $1$ because $p_1,\ldots,p_n$ are the densities of a probability mass function. Why is there a fake knife on the rack at the end of Knives Out (2019)? This course helped me to complete my final year project. Definition. $$ So, we will know that it's going to be snowy or cloudy. According to your expression: D K L ( u p) = log n 1 n x log p ( x) The entropy of p is H ( p) = x p ( x) log p ( x) But, there's no way to recover this from the first expression. This belongs to the category of maximum entropy probability distributions. Go Ahead BW. Hence, our measure must have high entropy for the first answer and lower one for the second. Are witnesses allowed to give private testimonies? Example. In this article we propose a new method based on uniform distributions and Cross- Entropy Clustering (CEC). And you're right in that entropy is maximal in a uniform distribution. Finally, entropy should be recursive with respect to independent events. You know that it's going to be one of those four weather conditions. This is a conservative measure. 1+\log(f(x))=c_0\cdot\color{#C00}{1}\tag5 So for two days, for two independent weather events, we know that there are 4 to the squared possible outcomes. It's a good question. /dev/random)? \int_a^bf(x)\,\mathrm{d}x=1\tag1 That is, the desired distribution is constant; that is, If he wanted control of the company, why didn't Elon Musk buy 51% of Twitter shares instead of 100%? H &= -\sum_{i=0}^{n-1} p_i \log p_i - (1-q)\log q\\ MIT, Apache, GNU, etc.) The best answers are voted up and rise to the top, Not the answer you're looking for? Isnt the entropy of the uniform distribution the maximum always? H*\ln 2 &= -\sum_{i=0}^{n-1} p_i \ln p_i - (1-q)\ln q H*\ln 2 &= -\sum_{i=0}^{n-1} p_i \ln p_i - (1-q)\ln q We can use $\delta f(x)=f(x)\,\delta\log(f(x))$ and use variations of $\log(x)$. The uniform must be least informative distribution, it's basically "I've no idea" answer. To learn more, see our tips on writing great answers. Measuring entropy of a uniform distribution source, Stop requiring only one assertion per unit test: Multiple assertions are fine, Going from engineer to entrepreneur takes more than just good code (Ep. The entropy is given by Claude Shannon in 1948. I understand this and that this corresponds to uniform distribution. We sketch the method in the next paragraph; see the section on general uniform distributions for more theory.. Final remark: an interesting aspect of entropy is that Shannon did not seem too bothered about the unicity of his definition. Explore Bachelors & Masters degrees, Advance your career with graduate-level learning. The min-entropy of a random variable is a lower bound on its entropy. Apply this for the concave function $f(x) = -x \log(x)$ and Jensen inequality for $y_i = p(x_i)$ and you have the proof. I.e.-, My greatest concern was what to call it. When there is an equal chance for all items to appear, we have a uniform distribution. 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